Tracheal branching in ants is area-decreasing, violating a central assumption of network transport models
Fig 3
Predictions and empirical data for the relationship between parent and child branch radii and their modelled consequences for CO2 efflux.
(a) Empirical outcomes (slopes ± standard error) of Ordinary Least Squares (OLS) regression forced through the intercept (sum of radii of child branchesα ~ radius of parent branchα where α = 1, 2 or 3 for Nunome’s, DaVinci’s or Murray’s laws, respectively; left hand points, solid lines) and Major Axis (Model II) regression (right hand points, dashed lines) for levels 1 to 2, and 2 to 3 (full results in S1 Table). Shaded boxes indicate upper and lower confidence intervals of each model fit and error bars indicate the standard error of the forced intercept models. A slope of approximately 1 is expected if the relationship fits observed data. (b) Relationship between parent and child tracheal branch radii for levels 1–2 (green dots) and 2–3 (blue dots), with the black fitted line indicating an ordinary least squares slope of ~1 (α = 1), whereas the yellow and orange lines indicate expectations for relationships with area-conserving (α = 2) and area-increasing (α = 3) branching, respectively. (c) Modelled whole-organism CO2 flux at each tracheal level (level 5 is tissue level, level 1 is spiracle) for area-conserving (yellow line), area-increasing (orange line), or area-reducing (black lines) transport networks. Each of the three black lines represents a model system with either no air sac (continuous solid line) or an air sac at either the 3rd or 4th level (indicated by the dashed line-segment), after which hydraulic resistance decreases and efflux is projected rather than modelled explicitly. Tracheal branch lengths of the modelled system ranged from 211 μm at the deepest level (level 5) to 1569 μm at level 1 (levels 4, 3, 2 had branch lengths of 352, 586, and 976 μm, respectively). The tracheal radius of level 5 was set at 3 μm. Whole-organism flux was modelled with a partial pressure difference of 6 kPa (see S1 Fig for 4 kPa and 8 kPa) and assuming 6 pairs of abdominal spiracles. Horizontal grey lines indicate the flux needed to meet the metabolic requirements of an average ant at rest or walking (S5 Table). Where flux values are below these activity lines, flux of the system is inferred to be insufficient to maintain activity by diffusion alone.